Place edamame in a large saucepan. Cover with water to 2 inches above edamame, and bring to a boil. Cook 10 minutes or until soft. Drain edamame, and set aside.
Place potato and chicken broth in a large saucepan; add enough water to cover potatoes. Bring to a boil. Reduce heat; simmer 15 minutes or until tender. Drain in a colander over a bowl, reserving 1/2 cup cooking liquid.
Place edamame in a food processor; process 1 minute or until finely chopped. With food processor on, slowly pour reserved cooking liquid through food chute; process until smooth.
Place edamame mixture, potato mixture, milk, oil, salt, and pepper in a large bowl, and mash with a potato masher to desired consistency.